82.23.9 problem Ex. 10
Internal
problem
ID
[18865]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
IV.
Singular
solutions.
problems
on
chapter
IV.
page
49
Problem
number
:
Ex.
10
Date
solved
:
Tuesday, January 28, 2025 at 12:31:12 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _Clairaut]
\begin{align*} {y^{\prime }}^{2} \left (-a^{2}+x^{2}\right )-2 x y y^{\prime }+y^{2}-b^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 82
dsolve(diff(y(x),x)^2*(x^2-a^2)-2*diff(y(x),x)*x*y(x)+y(x)^2-b^2=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {\sqrt {a^{2}-x^{2}}\, b}{a} \\
y \left (x \right ) &= -\frac {\sqrt {a^{2}-x^{2}}\, b}{a} \\
y \left (x \right ) &= c_{1} x -\sqrt {a^{2} c_{1}^{2}+b^{2}} \\
y \left (x \right ) &= c_{1} x +\sqrt {a^{2} c_{1}^{2}+b^{2}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 3.479 (sec). Leaf size: 419
DSolve[D[y[x],x]^2*(x^2-a^2)-2*D[y[x],x]*x*y[x]+y[x]^2-b^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [-\frac {\sqrt {a^2 \left (y(x)^2-b^2\right )} \arctan \left (\frac {\sqrt {y(x)^2-b^2}}{b}\right )}{b \sqrt {y(x)^2-b^2}}-\frac {2 \sqrt {y(x)^2-b^2} \sqrt {-a^2 \left (b^2-y(x)^2\right )} \arctan \left (\frac {b x \sqrt {y(x)^2-b^2}}{y(x) \left (\sqrt {a^2 \left (y(x)^2-b^2\right )}-\sqrt {a^2 \left (y(x)^2-b^2\right )+b^2 x^2}\right )+b^2 x}\right )}{b^3-b y(x)^2}&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {\sqrt {a^2 \left (y(x)^2-b^2\right )} \arctan \left (\frac {\sqrt {y(x)^2-b^2}}{b}\right )}{b \sqrt {y(x)^2-b^2}}+\frac {2 \sqrt {y(x)^2-b^2} \sqrt {-a^2 \left (b^2-y(x)^2\right )} \arctan \left (\frac {b x \sqrt {y(x)^2-b^2}}{y(x) \left (\sqrt {a^2 \left (y(x)^2-b^2\right )+b^2 x^2}-\sqrt {a^2 \left (y(x)^2-b^2\right )}\right )+b^2 x}\right )}{b^3-b y(x)^2}&=c_1,y(x)\right ] \\
y(x)\to -\frac {b \sqrt {a^2-x^2}}{a} \\
y(x)\to \frac {b \sqrt {a^2-x^2}}{a} \\
\end{align*}